Here we will discuss about one of the important geometrical property of parallelogram.
A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel
Given: PQRS is a quadrilateral in which PQ = SR and PQ ∥ SR.
To prove: PQRS is a parallelogram.
Construction: Join PR and QS such that they intersect at O.
Proof:
Statement |
Reason |
In ∆OPQ and ∆ORS, 1. ∠OPQ = ∠ORS |
1. PQ ∥ SR and PR is a transversal. |
2. ∠POQ = ∠ROS |
2. Opposite angles are equal. |
3. PQ = RS |
3. Given. |
4. ∆OPQ ≅ ∆ORS Therefore, OP = OR, OQ = OS. In ∆OPS and ∆OQR, |
4. By AAS criterion of congruency. CPCTC |
5. OP = OC, OQ = OS, ∠POS = ∠QOR |
5. By statement 4 and reason 2. |
6. ∆OPS ≅ ∆OQR Therefore, PS = QR, ∠OPS= ∠ORQ |
6. By SAS criterion of congruency. CPCTC |
7. PS ∥QR. |
7. Alternate angles are equal. |
8. PQRS is a parallelogram (Proved). |
8. PQ ∥ SR and statement 7. |
Corollary: In a parallelogram, each pair of opposite sides are parallel as well as equal.
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